Properties

Label 7056.2.k.b
Level $7056$
Weight $2$
Character orbit 7056.k
Analytic conductor $56.342$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.3424436662\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} +O(q^{10})\) \( q + \beta_{3} q^{5} -\beta_{1} q^{11} -3 \beta_{2} q^{13} + 2 \beta_{3} q^{17} -\beta_{2} q^{19} -4 \beta_{1} q^{23} + q^{25} -2 \beta_{1} q^{29} + \beta_{2} q^{31} - q^{37} -3 \beta_{3} q^{41} + q^{43} + 5 \beta_{3} q^{47} -2 \beta_{1} q^{53} -2 \beta_{2} q^{55} -2 \beta_{3} q^{59} -2 \beta_{2} q^{61} -9 \beta_{1} q^{65} -11 q^{67} + 5 \beta_{1} q^{71} -\beta_{2} q^{73} -5 q^{79} -3 \beta_{3} q^{83} + 12 q^{85} -2 \beta_{3} q^{89} -3 \beta_{1} q^{95} -6 \beta_{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{25} - 4q^{37} + 4q^{43} - 44q^{67} - 20q^{79} + 48q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
1.22474 0.707107i
0 0 0 −2.44949 0 0 0 0 0
881.2 0 0 0 −2.44949 0 0 0 0 0
881.3 0 0 0 2.44949 0 0 0 0 0
881.4 0 0 0 2.44949 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.k.b 4
3.b odd 2 1 inner 7056.2.k.b 4
4.b odd 2 1 441.2.c.a 4
7.b odd 2 1 inner 7056.2.k.b 4
7.c even 3 1 1008.2.bt.b 4
7.d odd 6 1 1008.2.bt.b 4
12.b even 2 1 441.2.c.a 4
21.c even 2 1 inner 7056.2.k.b 4
21.g even 6 1 1008.2.bt.b 4
21.h odd 6 1 1008.2.bt.b 4
28.d even 2 1 441.2.c.a 4
28.f even 6 1 63.2.p.a 4
28.f even 6 1 441.2.p.a 4
28.g odd 6 1 63.2.p.a 4
28.g odd 6 1 441.2.p.a 4
84.h odd 2 1 441.2.c.a 4
84.j odd 6 1 63.2.p.a 4
84.j odd 6 1 441.2.p.a 4
84.n even 6 1 63.2.p.a 4
84.n even 6 1 441.2.p.a 4
140.p odd 6 1 1575.2.bk.c 4
140.s even 6 1 1575.2.bk.c 4
140.w even 12 2 1575.2.bc.a 8
140.x odd 12 2 1575.2.bc.a 8
252.n even 6 1 567.2.s.d 4
252.o even 6 1 567.2.s.d 4
252.r odd 6 1 567.2.i.d 4
252.u odd 6 1 567.2.i.d 4
252.bb even 6 1 567.2.i.d 4
252.bj even 6 1 567.2.i.d 4
252.bl odd 6 1 567.2.s.d 4
252.bn odd 6 1 567.2.s.d 4
420.ba even 6 1 1575.2.bk.c 4
420.be odd 6 1 1575.2.bk.c 4
420.bp odd 12 2 1575.2.bc.a 8
420.br even 12 2 1575.2.bc.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 28.f even 6 1
63.2.p.a 4 28.g odd 6 1
63.2.p.a 4 84.j odd 6 1
63.2.p.a 4 84.n even 6 1
441.2.c.a 4 4.b odd 2 1
441.2.c.a 4 12.b even 2 1
441.2.c.a 4 28.d even 2 1
441.2.c.a 4 84.h odd 2 1
441.2.p.a 4 28.f even 6 1
441.2.p.a 4 28.g odd 6 1
441.2.p.a 4 84.j odd 6 1
441.2.p.a 4 84.n even 6 1
567.2.i.d 4 252.r odd 6 1
567.2.i.d 4 252.u odd 6 1
567.2.i.d 4 252.bb even 6 1
567.2.i.d 4 252.bj even 6 1
567.2.s.d 4 252.n even 6 1
567.2.s.d 4 252.o even 6 1
567.2.s.d 4 252.bl odd 6 1
567.2.s.d 4 252.bn odd 6 1
1008.2.bt.b 4 7.c even 3 1
1008.2.bt.b 4 7.d odd 6 1
1008.2.bt.b 4 21.g even 6 1
1008.2.bt.b 4 21.h odd 6 1
1575.2.bc.a 8 140.w even 12 2
1575.2.bc.a 8 140.x odd 12 2
1575.2.bc.a 8 420.bp odd 12 2
1575.2.bc.a 8 420.br even 12 2
1575.2.bk.c 4 140.p odd 6 1
1575.2.bk.c 4 140.s even 6 1
1575.2.bk.c 4 420.ba even 6 1
1575.2.bk.c 4 420.be odd 6 1
7056.2.k.b 4 1.a even 1 1 trivial
7056.2.k.b 4 3.b odd 2 1 inner
7056.2.k.b 4 7.b odd 2 1 inner
7056.2.k.b 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7056, [\chi])\):

\( T_{5}^{2} - 6 \)
\( T_{11}^{2} + 2 \)
\( T_{13}^{2} + 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -6 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 2 + T^{2} )^{2} \)
$13$ \( ( 27 + T^{2} )^{2} \)
$17$ \( ( -24 + T^{2} )^{2} \)
$19$ \( ( 3 + T^{2} )^{2} \)
$23$ \( ( 32 + T^{2} )^{2} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( 3 + T^{2} )^{2} \)
$37$ \( ( 1 + T )^{4} \)
$41$ \( ( -54 + T^{2} )^{2} \)
$43$ \( ( -1 + T )^{4} \)
$47$ \( ( -150 + T^{2} )^{2} \)
$53$ \( ( 8 + T^{2} )^{2} \)
$59$ \( ( -24 + T^{2} )^{2} \)
$61$ \( ( 12 + T^{2} )^{2} \)
$67$ \( ( 11 + T )^{4} \)
$71$ \( ( 50 + T^{2} )^{2} \)
$73$ \( ( 3 + T^{2} )^{2} \)
$79$ \( ( 5 + T )^{4} \)
$83$ \( ( -54 + T^{2} )^{2} \)
$89$ \( ( -24 + T^{2} )^{2} \)
$97$ \( ( 108 + T^{2} )^{2} \)
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